A356742 -id:A356742 - cp's OEIS frontend

A356744 Numbers k such that both k and k+2 have exactly 8 divisors.

40, 54, 102, 128, 136, 152, 182, 184, 230, 246, 248, 374, 424, 470, 472, 534, 582, 663, 710, 806, 822, 824, 854, 872, 902, 904, 999, 1105, 1192, 1256, 1309, 1334, 1336, 1432, 1446, 1526, 1542, 1545, 1576, 1593, 1645, 1686, 1784, 1832, 1864, 1885, 1910, 1928, 2006, 2013

Offset: 1

Jianing Song, Aug 25 2022

nonn

			54 is a term since 54 and 56 both have 8 divisors.

Jianing Song, Table of n, a(n) for n = 1..10000

Numbers k such that k and k+2 both have exactly m divisors: A001359 (m=2), A356742 (m=4), A356744 (m=6), this sequence (m=8).

Cf. also A274357 (numbers k such that k and k+1 both have exactly 8 divisors).

isA356744(n) = numdiv(n)==8 && numdiv(n+2)==8

A356743 Numbers k such that k and k+2 both have exactly 6 divisors.

Original entry on oeis.org

18, 50, 242, 243, 423, 475, 603, 637, 722, 845, 925, 1682, 1773, 2007, 2523, 2525, 2527, 3123, 3175, 3177, 4203, 4475, 4525, 4923, 5823, 6725, 6811, 6962, 7299, 7442, 7675, 8425, 8957, 8973, 9457, 9925, 10051, 10082, 10467, 11673, 11709, 12427, 12482, 12591, 13023, 13075

Offset: 1

Jianing Song, Aug 25 2022

nonn

If an even number has exactly 6 divisors, then it is of the form 32, 4*p or 2*p^2 for an odd prime p. Note that 4*p + 2 = 2*q^2 is impossible since q^2 - 1 is divisible by 24 for prime q >= 5. As a result, if k is an even term, then it is of the form 2*p^2 such that (p^2+1)/2 is a prime (p is in A048161).

			50 is a term since 50 and 52 both have 6 divisors.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A048161.
Numbers k such that k and k+2 both have exactly m divisors: A001359 (m=2), A356742 (m=4), this sequence (m=6), A356744 (m=8).
Cf. also A049103 (numbers k such that k and k+1 both have exactly 6 divisors).

isA356743(n) = numdiv(n)==6 && numdiv(n+2)==6

A356766 Least number k such that k and k+2 both have exactly 2n divisors, or -1 if no such number exists.

Original entry on oeis.org

3, 6, 18, 40, 127251, 198, 26890623, 918, 17298, 6640, 25269208984375, 3400, 3900566650390623, 640062, 8418573, 18088, 1164385682220458984373, 41650, 69528379848480224609373, 128464, 34084859373, 12164094, 150509919493198394775390625, 90270, 418514293125, 64505245696

Offset: 1

Jean-Marc Rebert, Aug 26 2022

nonn

			For n=1, numdiv(3) = numdiv(5) = 2 = 2*1, and no number < 3 satisfies this, hence a(1) = 3.

Numbers k such that k and k+2 both have exactly m divisors: A001359 (m=2), A356742 (m=4), A356743 (m=6), A356744 (m=8).

Cf. A000005 (d(n)), A003680, A005238, A006558, A006601, A062832, A067888, A067889, A075036.

a={}; n=1; nmax=10; For[k=1, n<=nmax, k++, If[DivisorSigma[0, k] == DivisorSigma[0, k+2] == 2n, AppendTo[a, k]; k=1; n++]]; a (* Stefano Spezia, Aug 26 2022 *)
Flatten[Table[SequencePosition[DivisorSigma[0,Range[27*10^6]],{2n,,2n},1],{n,10}],1][[;;,1]] (* The program generates the first 10 terms of the sequence. To generate more, increase the Range constant but the program will take a long time to run. *) (* _Harvey P. Dale, Jul 01 2023 *)

a(n)=for(k=1,+oo,if(numdiv(k)==2*n&&numdiv(k+2)==2*n,return(k)))

More terms from Jinyuan Wang, Aug 28 2022

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A356744 Numbers k such that both k and k+2 have exactly 8 divisors.

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A356743 Numbers k such that k and k+2 both have exactly 6 divisors.

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A356766 Least number k such that k and k+2 both have exactly 2n divisors, or -1 if no such number exists.

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